A useful density lemma is the following.

Let $(X,\mathcal{A}, \mu)$ be a
measure space and let $\Gamma\subset
> \mathcal{P}(X)$ a ring of sets
of finite measure that generates the
$\sigma$-algebra $\mathcal{A}$. Then,
the linear span of the characteristic
functions of sets in $\Gamma$ is dense
in $L^p$ (here $1\le p < +\infty$)

In your case, of course, you can take $\Gamma$ to be the collection of finite unions of Cartesian products of measurable sets of finite measure.

Incidentally, the hypothesis on $\Gamma$ in that density lemma can be weakened (one does not need all the ring strucure of $\Gamma$). Let's say that $\Gamma\subset \mathcal{P}(X)$ is a "semi-ring" (*warning*: not standard; I borrowed it from Halmos, with a slightly more general meaning) if the following holds:

For all $A$ and $B$ in $\Gamma$ the
sets $A\setminus B$ and $A\cap B$ are
both expressible as union of
countably many disjoint element of
$\Gamma$.

Then the above lemma holds true. The notion of "semi-ring"is also interesting, in that it is a convenient domain for a completely additive set function, in order that the Caratheodory's Extension Theorem holds.